Abstract

Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.

We are grateful to O. Gürcan for useful discussions and to our colleagues of the Jean Lamour institute in Nancy and of the IRFM institute at Cadarache, for having introduced us to the physics of tokamaks. We are also grateful to U. Frisch for many remarks on an early version of this manuscript. This work was supported by the VLASIX and EUROFUSION projects, respectively under Grant Nos. ANR-13-MONU-0003-01 and EURATOM-WP15-ENR-01/IPP-01.

Article outline:I. INTRODUCTIONA. Motivations and key issuesB. Presentation and explanation of the resultsC. Advantages and drawbacks of the asymptotic approachD. Organisation of the paperII. THE GYROKINETIC FRAMEWORKA. The gyrokinetic-vlasov equationB. The gyrokinetic-waterbag model1. The waterbag reduction concept in 1D2. The gyrokinetic-waterbag equationsC. The magnetic field line geometry and the toroidal coordinate systemD. Definition of scales and their orderingIII. DERIVATION OF THE EIGENVALUE PROBLEM FOR THE GYROKINETIC-WATERBAG MODELA. Linearization of the gyrowaterbag modelB. The analytic representation of the steady equilibrium stateC. System for the perturbation1. Field-aligned coordinate system2. Ballooning transformation and eikonal representation3. The well-suited system for the perturbationIV. ASYMPTOTIC ANALYSISA. Asymptotic expansionB. The zeroth-order systemC. The first-order systemD. The second-order systemE. Asymptotic analysis including a non-trivial gyroaverage operatorF. An algorithm for solving the eigenvalue problemV. SPECTRAL ANALYSISA. The schrödinger-type radial envelope equation1. General case2. Case of two closely spaced simple turning pointsB. The nested Fredholm-type integral operator1. Basic properties of the integral operator2. Case of only open contours3. Case of open and closed contours